On May 31, 1:02 am, chaja...@mail.com wrote:> On May 31, 12:33 am, William Hughes <wpihug...@hotmail.com> wrote:>> > let a = 0.999... be a real number. We do not need> > to give a full definition at this point> > a<=1> > and> > a>(1-(1/10^n) for any natural number n>> You have defined 0.999... to be a real number without jusitification.> I can make no such assumption. Each position within 0.999... can be> expressed as a real number, but the totality, the very infinite nature> of it, seems to render it a never ending relation more than a specific> explicit location on the real number line.

Oh, I'm SO glad you said this. I think the same thing! In fact, itgeneralizes to other digits besides 9 as well! The case with0 is particularly interesting. Clearly 0.000... is also a "neverending relation more than a specific location on the realnumber line." I mean, *obviously* it's different from 0. Lookat all those additional digits on the end!

Marshall

PS. Sadly, because this is sci.math, I have to be explicit thatI am speaking ironically.